Monday, February 20, 2012

Who says energy comes in lumps?

Everyone says energy comes in lumps. That's the basic starting point of our whole paradigm. You'd have to be crazy to argue with it.

Call me crazy, but where are the lumps? The lumps are supposed to be necessary to derive the black-body spectrum, but just where do they come from? When we left off the other day, I was contrasting the Copenhagen interpretation with the Schroedinger interpretation. In Copenhagen, the atoms in a hot gas are each in some particular state of definite energy. According to certain probabilities, they may randomly absorb or emit a quantum of energy, thereby jumping from one state to another. Energy always appears in discrete lumps, or quanta.

In the Schroedinger picture, atom is surrounded by a charge of cloud which can have any shape at all. From our human perspective it is convenient for us to analyze those shapes as the sum of what are called eigenstates: those correspond to the definite energy states of Copenhagen. The equations show that any combination of eigenstates will generate an oscillating charge distribution, which will tend to radiate or absorb energy, depending on the state of the ambient electromagnetic field. The atoms are doing this all the time except in the exceptional circumstance when they happen to find themselves in a pure eigenstate. Only then does its charge distribution become stationary, and accordingly it stops exchanging energy with the surrounding field, if only for a moment.

How can these two radically different pictures give the same physics? The obvious answer is that they don't. Everyone knows that energy is absorbed or given off by atoms only in discrete quanta. The so-called Schroedinger picture, as I've described it, violates this fundamental rule. Therefore it must be wrong.

But why is it wrong? If it's so wrong, it should be easy to show where it falls down. Let's take a typical calculation. But what exactly should we calculate? It's easier said than done.

In situations like this, the thing to do is to back right down to the simplest possible case. In the Copenhagen Theory, you have something called the Einstein A and B coefficients. They give you the probability of an atom jumping from one state to another state. The Einstein A coefficitent has a particularly simple interpretation: it represesents an atom in an excited state which emits a photon and jumps to the ground state. For the s-p transition of hydrogen, we can actually calculate it. (Or we can just look it up.)

What does the Schroedinger picture tell us about this transition? Well, on the face of it, things don't look so good for Schroedinger. An atom in the pure p-state has a stationary electron cloud, so it doesn't radiate at all. It stays that way forever. This clearly contradicts what we know to be true from Copenhagen.

Or does it? Just how do we put an atom into a pure p-state, and how long do we have to watch it before it decays into the s-state? It's not obvious how we would do an experiment to verify the Copenhagen results. And yet we know it to be true. How do we know it? Because we take bulk measurements of the radiation intensity of a gas containing atoms in both states. Knowing the total energy of the gas, we know what percentage are in the excited state. Copenhagen predicts how often they decay, and knowing the amount of energy in each decay, we can calculate the average radiation. The experiment verifies that Copenhagen is correct.

But notice that this experiment doesn't tell us anything about individual atoms suddenly jumping from one state to another. That is a human spin that Copenhagen attaches to the physics. To compare the Schroedinger result, we ought to compare only the bottom-line physics. That means we have a gas with a certain amount of energy distributed between the ground state and the excited state, and there is a certain amount of electromagnetic energy associated with that gas. That's what we can measure, and that's what we have to calculate using the Schroedinger picture.

Let's say that 10% of the atoms are in the excited state. Of course, that's Copenhagen language: Schroedinger would have us say that each of the atoms is in a partially excited state, to the extent of 10%. Because of the mix between the s and p states, each atom has an oscillating charge distribution. This means it is a tiny antenna. The laws for calculating radiation from antennas are well known. We just need to figure out how much the atoms are oscillating, and apply the antenna equations.

Do you think we will get the same result as the Copenhagen calculation?